Devlin's
Angle

June 2003

When is a proof?

What is a proof? The question has two answers.
The right wing ("right-or-wrong", "rule-of-law")
definition is that a proof is a logically correct
argument that establishes the truth of a given
statement. The left wing answer (fuzzy, democratic,
and human centered) is that a proof is an argument
that convinces a typical mathematician of the truth
of a given statement.

While valid in an idealistic sense, the right wing
definition of a proof has the problem that, except for
trivial examples, it is not clear that anyone has ever
seen such a thing. The traditional examples of correct
proofs that have been presented to students for
over two thousand years are the geometric arguments
Euclid presents in his classic text Elements,
written around 350 B.C. But as Hilbert pointed out
in the late 19th century, many of those arguments
are logically incorrect. Euclid made repeated use
of axioms that he had not stated, without which his
arguments are not logically valid.

Well, can we be sure that the post-Hilbert versions
of Euclid's arguments are right wing proofs? Like most
mathematicians, I would say yes. On what grounds do
I make this assertion? Because those arguments
convince me and have convinced all other mathematicians
I know. But wait a minute, that's the left
wing definition of proof, not the right wing one.

And there you have the problem. Like right wing
policies, for all that it appeals to individuals
who crave certitude in life, the right wing definition
of mathematical proof is an unrealistic ideal that does
not survive the first contact with the real world.
(Unless you have an army to impose it with force, an
approach that mathematicians have hitherto shied
away from.)

Even in the otherwise totally idealistic (and right
wing) field of mathematics, the central notion of
proof turns out to be decidedly left wing the
moment you put it to work. In other words, the only
notion of proof that makes real sense in mathematics,
and which applies to what mathematicians actually do,
is the left wing notion.

So much for "What is a proof?" An argument becomes a
proof when the mathematical community agrees it is
such. How then about the related question "When is
proof?" At what point does the community of
mathematicians agree that a purported statement has
been proved? When does the argument presented become
a proof?

In last month's column, I looked at three recent
examples of mathematical proofs that illustrate this
question "When is proof?" They are: Andrew Wiles'
1993 presentation of a proof of Fermat's Last Theorem,
Russian mathematician Grigori Perelman's 2002 claim
to have "possibly proved" the Poincare Conjecture,
and Daniel Goldston and Cem Yildirim's 2003
announcement of a major advance on the Twin Primes
Conjecture. All three are far too long and complicated
for anyone to seriously believe these are anything
but left wing proofs.

In Wiles' case, a major flaw was discovered in his
argument within months of his initial announcement,
which took him over a year to fix. (His proof
is now accepted as being correct.) Perelman has been
guarded in his claim, admitting that it will likely
take months before the mathematical community will
know for sure whether he is right or not. In the case
of the Goldston-Yildirim result, they and the rest of
the mathematical community were still sipping their
celebratory champagne when Andrew Granville of the
University of Montreal and Kannan Soundarajan of the
University of Michigan discovered a flaw in the new
proof, a flaw that is almost certainly fatal.

Subsequent to the publication of my commentary,
Granville sent me an email providing some background
on what led him and his colleague to discover the
error in the Goldston-Yildirim proof. This episode
provides an excellent example of the psychology of
doing mathematics.

Like everyone else, when they first read through
Goldston and Yildirim's proof, Granville and
Soundarajan thought it was correct. The result
was dramatic and surprising, but the argument
seemed to work. Like any new mathematical result,
the proof was a mixture of old and familiar techniques
and some new ideas. Because the result was such a
major breakthrough, Granville and Soundarajan, like
everyone else, looked especially carefully at the
new ideas. Everything seemed okay.

Goldston and Yildirim did not claim to have proved
the Twin Prime Conjecture, that there are infinitely
many pairs of primes just 2 apart (such as 3 and 5 or
11 and 13), but they did claim to have made major
progress in that direction, showing that there are
infinitely many primes p such that the gap
to the next prime is very small. (See last month's
column for a precise statement.)

Granville and Soundarajan took Goldston and Yildirim's
argument and extended it to show that there are
infinitely many pairs of primes differing by no more
than 12. In Granville's own words, "We were damn close
to twin primes!"

Too close, in fact. Not believing their result, the two
decided to look again at Goldston and Yildirim's core
lemmas to see if there was some crucial detail that was
being too easily glossed over. It took only a couple of
hours to home in on one tiny detail that was not fully
explained and which they could not prove. And with the
discovery of that one tiny flaw, buried deep in the
"tried and tested" part of the argument that
everyone had accepted as correct, the entire Goldston
and Yildirim result fell apart. It wasn't that the
established procedure was in itself wrong; rather, it
did not apply under the new circumstances in which
Goldston and Yildirim were using it.

Brian Conrey, the director of the American Institute of
Mathematics, which played an instrumental role in the
research that led Goldston and Yildirim to develop their
argument, has commented on how this incident highlights
the psychology of breakthrough in mathematics.
Goldston and Yildirim's core lemmas had a familiar
flavor and their conclusions were very believable, so
everyone believed them. The expectation was that if
there was a mistake it would be among the new ideas.
So, once several people (including Granville and
Soundarajan) had verified that the new ideas were
correct, everyone signed off on the new result as being
correct. Granville himself arranged to give a series of
lectures on the new proof.

But then he and Soundarajan made their own, "gap 12"
discovery. As Granville puts it, this took them from the
"fantastic" (Goldston and Yildirim's result) into the
realm of "unbelievable". At that point the psychology
changed. Neither Granville nor Soundarajan really
believed it could possibly be correct. With that change
in belief it became relatively straightforward to
pinpoint the error.

But as Granville himself points out, the psychology
is important here. They had to have good reason to
suspect there was an error before they were able to
find it. Conrey has observed that if Granville and
Soundarajan had not used the new method to make their
own "unbelievable" gap-12 deduction, the
Goldston-Yildirim proof would in all probability have
been published and the mistake likely not found for
some years.

It makes you think, doesn't it?

It made me wonder about the true status of another
highly problematic recent breakthrough in mathematics,
University of Michigan mathematician Thomas Hales'
1998 announcement that after six years of effort he
had finally found a proof of Kepler's Sphere Packing
Conjecture.

The problem began with a guess - we can't really call
it more than that - Johannes Kepler made back in 1611
about the most efficient way to pack equal-sized
spheres together in a large crate. Is the most efficient
packing (i.e. the one that packs most spheres into a
given sized crate) the one where the spheres lie in
staggered layers, the way greengrocers the world
over stack oranges, so that the oranges in each higher
layer sit in the hollows made by the four oranges
beneath them? (The formal term for this orange-pile
arrangement is a face-centered cubic lattice.)

For a small crate, the answer can depend on the
actual dimensions of the crates and the spheres. But
for a very large crate, you can show, as Kepler did,
that the orange-pile arrangement is always more
efficient than a number of other regular arrangements.
But was it the world champion?

The general problem as considered by Kepler and
subsequent mathematicians is formulated not in terms
of the number of spheres that can be packed together
but the density of the packing, i.e., the total
volume of the spheres divided by the total volume of
the container into which they are packed. The problem
is further generalized by defining the density of
a packing (pattern) as the limit of the densities
of individual packings (using that pattern) for cubic
crates as the volume of the crates approaches infinity.

According to this definition, the orange-pile packing
has a density of pi/3sqrt(2) (approximately 0.74).
Kepler believed that this is the densest of all
arrangements, but was unable to prove it. So were
countless succeeding generations of mathematicians.

(In 1993, a highly-respected mathematician at the
University of California at Berkeley produced a
complicated proof of the Kepler Conjecture which,
after several years of debate, most mathematicians
decided was incorrect.)

Major progress on the problem was made in the 19th
century, when the legendary German mathematician
and physicist Karl Friedrich Gauss managed to prove
that the orange-pile arrangement was the most
efficient among all "lattice packings." A lattice
packing is one where the centers of the spheres are
all arranged in a "lattice", a regular three-dimensional
grid, like a lattice fence.

But there are non lattice arrangements that are almost
as efficient than the orange-pile, so Gauss's result
did not solve the problem completely.

The next major advance came in 1953, when a
Hungarian mathematician, Laszlo Fejes Toth, managed
to reduce the problem to a huge calculation involving
many specific cases. This opened the door to solving
the problem using a computer.

In 1998, after six years work, Hales announced that
he had indeed found a computer-aided proof. He posted
the entire argument on the Internet. The proof involved
hundreds of pages of text and gigabytes of computer
programs and data. To "follow" Hales' argument, you
had to download his programs and run them.

Hales admitted at the time that, with a proof this
long and complex, involving a great deal of computation,
it would be some time before anyone could be absolutely
sure it is correct. By posting everything on the world
wide web, he was challenging the entire mathematical
community to see if they could find anything wrong.

Hales' result was so important that, soon after he made
his announcement, the highly prestigious journal Annals
of Mathematics made the unusual step of actively
soliciting the paper for publication, and hosted a
conference in January 1999 that was devoted to
understanding the proof. A panel of 12 referees was
assigned to the task of verifying the correctness of
the proof, with world expert Toth in charge of the
reviewing process.

After four full years of deliberation, Toth returned a
report stating that he was 99% certain of the correctness
of the proof, but that the team had been unable to
completely certify the argument.

In a letter to Hales, Robert MacPherson, the editor of
the journal, said of the evaluation process:

The news from the referees is bad, from my perspective.
They have not been able to certify the correctness of
the proof, and will not be able to certify it in the
future, because they have run out of energy to devote
to the problem. This is not what I had hoped for.

The referees put a level of energy into this that is,
in my experience, unprecedented. They ran a seminar on
it for a long time. A number of people were involved,
and they worked hard. They checked many local statements
in the proof, and each time they found that what you
claimed was in fact correct. Some of these local checks
were highly non-obvious at first, and required weeks to
see that they worked out. The fact that some of these
worked out is the basis for the 99% statement of Fejes
Toth.

Well, how far have we come in ruling on this proof, five
years after it was first announced? Experts who have
visited Hales' website and looked through the material
have said that it looks convincing. But no one has yet
declared it to be 100% correct. And with the recent
episode of Goldston and Yildirim's incomparably less
complicated argument about prime numbers still fresh
in our minds, would we be prepared to sign off on Hales'
result even if someone had made such a claim?

Hales himself sees the process of verifying his proof
as an active work in progress. He has initiated what he
calls the Flyspeck Project, the goal of which is to
produce a more detailed (and hence more right wing)
proof of the Kepler Conjecture. (He came up with the
name 'flyspeck' by matching the pattern /f.*p.*k/
against a English dictionary. FPK in turn is an
acronym for "The Formal Proof of Kepler." The word
'flyspeck' can mean to examine closely or in minute
detail; or to scrutinize. As Hales observes, the term is
highly appropriate for a project intended to scrutinize
the minute details of a mathematical proof.)

Here is how Hales describes what he means by the term
"formal proof" in his project title.

Traditional mathematical proofs are written in a way to
make them easily understood by mathematicians. Routine
logical steps are omitted. An enormous amount of context
is assumed on the part of the reader. Proofs, especially
in topology and geometry, rely on intuitive arguments in
situations where a trained mathematician would be capable
of translating those intuitive arguments into a more
rigorous argument.

In a formal proof, all the intermediate logical steps are
supplied. No appeal is made to intuition, even if the
translation from intuition to logic is routine. Thus, a
formal proof is less intuitive, and yet less susceptible
to logical errors.

Clearly, what Hales is talking about here is something
that could only be carried out on a computer running
purposely written software.

Hence the Flyspeck Project, which Hales believes is the
only way to tackle the problem of verifying a proof
such as his. The idea is to make use of two resources
that were not available to previous generations of
mathematicians: the Internet and massive amounts of
computer power. "It is not the sort of project that can
be completed by a single individual" Hales says. "Instead
it will require the collective efforts of a large and
dedicated team over a long period."

He is currently recruiting collaborators and team members
from around the world to work with him on the project.
He estimates that it may take 20 work-years to complete
the task. At the end of which, the mathematical community
may indeed be able to declare Kepler's Conjecture as
finally proven. But what will such a statement really
mean? The computer proof will, I think we will all agree,
be a right wing proof. Of something. But then there
is the thorny question of deciding whether what the computer
has done amounts to a proof of the Kepler Conjecture.
And that is a decision that only the mathematical community
can make. We will have to decide that the computer program
really does what its designers intended, and whether that
intention does in fact prove the Conjecture. And those
parts of the process are inescapably left wing. In other
words, the Flyspeck Project amounts to making use of the
process of generating a right wing proof as a method
for arriving at a left wing proof.

Toth thinks that this situation will occur more
and more often in mathematics. He says it is similar
to the situation in experimental science - other
scientists acting as referees cannot certify the
correctness of an experiment, they can only subject
the paper to consistency checks. He thinks that the
mathematical community will have to get used to this
state of affairs.

When it comes down to it, mathematics, for all that it
appears to be the most right wing of disciplines, turns
out in practice to be left wing to the core.

For more details on Hales' proof of Kepler's Conjecture
and on the Flyspeck Project, point your browser to the
Flyspeck
Project webpage.

For details on Perelman's work on the Poincare Conjecture,
click
here
and
here.

For details of the Goldston-Yildirim work, including a
discussion of the error, click
here.